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Properties

Label 1900.4.a.l
Level $1900$
Weight $4$
Character orbit 1900.a
Self dual yes
Analytic conductor $112.104$
Analytic rank $1$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1900,4,Mod(1,1900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1900.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1900.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(112.103629011\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 158x^{10} + 9117x^{8} - 255920x^{6} + 3734252x^{4} - 26837136x^{2} + 73273600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{11} q^{7} + (\beta_{2} - 1) q^{9} + (\beta_{5} - 9) q^{11} + (\beta_{10} - \beta_{9} - \beta_{7}) q^{13} + (2 \beta_{11} - 2 \beta_{10} + \cdots - \beta_1) q^{17} + 19 q^{19}+ \cdots + (2 \beta_{6} - 9 \beta_{5} + \cdots - 192) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{9} - 110 q^{11} + 228 q^{19} - 8 q^{21} + 76 q^{29} + 80 q^{31} - 60 q^{39} - 648 q^{41} + 886 q^{49} - 328 q^{51} - 984 q^{59} - 1970 q^{61} - 364 q^{69} - 3060 q^{71} - 552 q^{79} - 3388 q^{81}+ \cdots - 2366 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 158x^{10} + 9117x^{8} - 255920x^{6} + 3734252x^{4} - 26837136x^{2} + 73273600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61\nu^{10} - 8674\nu^{8} + 437273\nu^{6} - 10868196\nu^{4} + 131877836\nu^{2} - 574347568 ) / 2222064 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 605\nu^{10} - 78440\nu^{8} + 3248623\nu^{6} - 57488088\nu^{4} + 443719372\nu^{2} - 1222414820 ) / 555516 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 463\nu^{10} - 63502\nu^{8} + 2898659\nu^{6} - 58210740\nu^{4} + 520024100\nu^{2} - 1631938000 ) / 170928 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3587\nu^{10} - 491846\nu^{8} + 22434631\nu^{6} - 449918700\nu^{4} + 4022546596\nu^{2} - 12753359288 ) / 1111032 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 85289 \nu^{11} + 10755722 \nu^{9} - 423294253 \nu^{7} + 6987290220 \nu^{5} + \cdots + 114707836544 \nu ) / 2377608480 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 107881 \nu^{11} + 15482998 \nu^{9} - 767907557 \nu^{7} + 17579731320 \nu^{5} + \cdots + 805445727856 \nu ) / 2377608480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 65314 \nu^{11} + 9307927 \nu^{9} - 451609448 \nu^{7} + 9760187235 \nu^{5} + \cdots + 315187863964 \nu ) / 594402120 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 97289 \nu^{11} + 13547312 \nu^{9} - 632439373 \nu^{7} + 12994700190 \nu^{5} + \cdots + 389805074264 \nu ) / 594402120 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 470161 \nu^{11} - 64592308 \nu^{9} + 2956848017 \nu^{7} - 59609142030 \nu^{5} + \cdots - 1693350797416 \nu ) / 1188804240 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 26 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + 2\beta_{10} + 3\beta_{9} - \beta_{8} + 5\beta_{7} + 44\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{6} + 6\beta_{5} - 4\beta_{3} + 86\beta_{2} + 1093 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 183\beta_{11} + 197\beta_{10} + 251\beta_{9} - 76\beta_{8} + 446\beta_{7} + 2610\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -515\beta_{6} + 616\beta_{5} + 2\beta_{4} - 294\beta_{3} + 6342\beta_{2} + 62969 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13813\beta_{11} + 15381\beta_{10} + 18203\beta_{9} - 5434\beta_{8} + 33224\beta_{7} + 174302\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -40709\beta_{6} + 48564\beta_{5} + 360\beta_{4} - 18550\beta_{3} + 453282\beta_{2} + 4156301 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 995843\beta_{11} + 1124773\beta_{10} + 1291643\beta_{9} - 386528\beta_{8} + 2379562\beta_{7} + 12107254\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2987791\beta_{6} + 3558904\beta_{5} + 36854\beta_{4} - 1206478\beta_{3} + 32153602\beta_{2} + 287566777 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 70869089 \beta_{11} + 80493213 \beta_{10} + 91378567 \beta_{9} - 27425074 \beta_{8} + \cdots + 851559726 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.41136
−5.00165
−4.64681
−4.44306
−3.73144
−2.64108
2.64108
3.73144
4.44306
4.64681
5.00165
8.41136
0 −8.41136 0 0 0 11.8650 0 43.7510 0
1.2 0 −5.00165 0 0 0 −23.7493 0 −1.98350 0
1.3 0 −4.64681 0 0 0 26.3134 0 −5.40719 0
1.4 0 −4.44306 0 0 0 −30.8233 0 −7.25924 0
1.5 0 −3.73144 0 0 0 12.0773 0 −13.0764 0
1.6 0 −2.64108 0 0 0 −2.80379 0 −20.0247 0
1.7 0 2.64108 0 0 0 2.80379 0 −20.0247 0
1.8 0 3.73144 0 0 0 −12.0773 0 −13.0764 0
1.9 0 4.44306 0 0 0 30.8233 0 −7.25924 0
1.10 0 4.64681 0 0 0 −26.3134 0 −5.40719 0
1.11 0 5.00165 0 0 0 23.7493 0 −1.98350 0
1.12 0 8.41136 0 0 0 −11.8650 0 43.7510 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(19\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.4.a.l 12
5.b even 2 1 inner 1900.4.a.l 12
5.c odd 4 2 380.4.c.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.4.c.a ✓ 12 5.c odd 4 2
1900.4.a.l 12 1.a even 1 1 trivial
1900.4.a.l 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 158T_{3}^{10} + 9117T_{3}^{8} - 255920T_{3}^{6} + 3734252T_{3}^{4} - 26837136T_{3}^{2} + 73273600 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1900))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 158 T^{10} + \cdots + 73273600 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 59893606897216 \) Copy content Toggle raw display
$11$ \( (T^{6} + 55 T^{5} + \cdots - 328352864)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 53\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T - 19)^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 59\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 8230949770576)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 1268789878784)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 3754110924512)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 85\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 56\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 1097594995072)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 24\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 208576706426240)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 19\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 68\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
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